Fixed Point Strategies in Data Science

Combettes, Patrick L.; Pesquet, Jean-Christophe

doi:10.1109/tsp.2021.3069677

Cited by 74 publications

(45 citation statements)

References 200 publications

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“…Compositions of projectors are often employed in projection methods. This is a vast area which we will not summarize here; however, we refer the reader to [16] as a starting point as well as the very recent paper [18].…”

Section: Fixed Points Of Compositions Of Projectorsmentioning

confidence: 99%

“…This form of (Id −R) −1 makes it clear that this operator is strongly monotone with constant 1 2 which implies that the dual problem (18) has at most one solution (26) which is consistent with (19). (A feature of Attouch-Théra duality is that either both primal and dual have solutions or they both don't.…”

Section: Product Space and Attouch-théra Dualitymentioning

confidence: 99%

“…where the equality follows from ( 23) and (40). Next, y might even solve the original dual (18) in which case Z is given by (20). Whether or not this is the case, we always have, using (25), (40), and (39),…”

Section: The Proof Of the Geometry Conjecturementioning

confidence: 99%

See 2 more Smart Citations

The difference vectors for convex sets and a resolution of the geometry conjecture

Alwadani¹,

Bauschke²,

Revalski³

et al. 2021

Open Journal of Mathematical Optimization

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The geometry conjecture, which was posed nearly a quarter of a century ago, states that the fixed point set of the composition of projectors onto nonempty closed convex sets in Hilbert space is actually equal to the intersection of certain translations of the underlying sets.In this paper, we provide a complete resolution of the geometry conjecture. Our proof relies on monotone operator theory. We revisit previously known results and provide various illustrative examples. Comments on the numerical computation of the quantities involved are also presented.

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Section: Fixed Points Of Compositions Of Projectorsmentioning

confidence: 99%

Section: Product Space and Attouch-théra Dualitymentioning

confidence: 99%

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The difference vectors for convex sets and a resolution of the geometry conjecture

Alwadani¹,

Bauschke²,

Revalski³

et al. 2021

Open Journal of Mathematical Optimization

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show abstract

“…Moreover, function g belongs to

. We thus propose to make use of the proximal primal-dual algorithm ([ 40 , 41 ]) from Condat-Vú [ 42 , 43 ], which leads to Algorithm 1. Algorithm 1: Condat-Vú algorithm

…”

Section: Problem Statementmentioning

confidence: 99%

Calibration-Less Multi-Coil Compressed Sensing Magnetic Resonance Image Reconstruction Based on OSCAR Regularization

et al. 2021

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Over the last decade, the combination of compressed sensing (CS) with acquisition over multiple receiver coils in magnetic resonance imaging (MRI) has allowed the emergence of faster scans while maintaining a good signal-to-noise ratio (SNR). Self-calibrating techniques, such as ESPiRIT, have become the standard approach to estimating the coil sensitivity maps prior to the reconstruction stage. In this work, we proceed differently and introduce a new calibration-less multi-coil CS reconstruction method. Calibration-less techniques no longer require the prior extraction of sensitivity maps to perform multi-coil image reconstruction but usually alternate estimation sensitivity map estimation and image reconstruction. Here, to get rid of the nonconvexity of the latter approach we reconstruct as many MR images as the number of coils. To compensate for the ill-posedness of this inverse problem, we leverage structured sparsity of the multi-coil images in a wavelet transform domain while adapting to variations in SNR across coils owing to the OSCAR (octagonal shrinkage and clustering algorithm for regression) regularization. Coil-specific complex-valued MR images are thus obtained by minimizing a convex but nonsmooth objective function using the proximal primal-dual Condat-Vù algorithm. Comparison and validation on retrospective Cartesian and non-Cartesian studies based on the Brain fastMRI data set demonstrate that the proposed reconstruction method outperforms the state-of-the-art (ℓ1-ESPIRiT, calibration-less AC-LORAKS and CaLM methods) significantly on magnitude images for the T1 and FLAIR contrasts. Additionally, further validation operated on 8 to 20-fold prospectively accelerated high-resolution ex vivo human brain MRI data collected at 7 Tesla confirms the retrospective results. Overall, OSCAR-based regularization preserves phase information more accurately (both visually and quantitatively) compared to other approaches, an asset that can only be assessed on real prospective experiments.

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“…This is not a survey on solution algorithms for optimization problems and variational inequalities. Some relevant references on iterative methods include [23,13,58,1,59,60] and the references therein.…”

Section: What This Survey Is Not Aboutmentioning

confidence: 99%

Convergence of sequences: a survey

Franci¹,

Grammatico²

2021

Preprint

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Convergent sequences of real numbers play a fundamental role in many different problems in system theory, e.g., in Lyapunov stability analysis, as well as in optimization theory and computational game theory. In this survey, we provide an overview of the literature on convergence theorems and their connection with Féjer monotonicity in the deterministic and stochastic settings, and we show how to exploit these results.

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Fixed Point Strategies in Data Science

Cited by 74 publications

References 200 publications

The difference vectors for convex sets and a resolution of the geometry conjecture

The difference vectors for convex sets and a resolution of the geometry conjecture

Calibration-Less Multi-Coil Compressed Sensing Magnetic Resonance Image Reconstruction Based on OSCAR Regularization

Convergence of sequences: a survey

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